According to standard image reconstruction theories, in order to reconstruct an image without aliasing artifacts, the sampling rate employed to acquire image data must satisfy the so-called Nyquist criterion, which is set forth in the Nyquist-Shannon sampling theorem. Moreover, in standard image reconstruction theories, no specific prior information about the image is needed. On the other hand, when some prior information about the desired or target image is available and appropriately incorporated into the image reconstruction procedure, an image can be accurately reconstructed even if the Nyquist criterion is violated. For example, if one knows that a desired target image consists of only a single point, then only two orthogonal projections that intersect at that point are needed to accurately reconstruct the image point.
If prior information is known about the desired target image, such as if the desired target image is a set of sparsely distributed points, it can be reconstructed from a set of data that was acquired in a manner that does not satisfy the Nyquist criterion. Put more generally, knowledge about the sparsity of the desired target image can be employed to relax the Nyquist criterion; however, it is a highly nontrivial task to generalize these arguments to formulate a rigorous image reconstruction theory.
The Nyquist criterion serves as one of the paramount foundations of the field of information science. However, it also plays a pivotal role in modern medical imaging modalities such as magnetic resonance imaging (“MRI”) and x-ray computed tomography (“CT”). When the number of data samples acquired by an imaging system is less than the requirement imposed by the Nyquist criterion, artifacts appear in the reconstructed images. In general, such image artifacts include aliasing and streaking artifacts. In practice, the Nyquist criterion is often violated, whether intentionally or through unavoidable circumstances. For example, in order to shorten the data acquisition time in a time-resolved MR angiography study, image data is often acquired using a radial sampling pattern that undersamples the peripheral portions of k-space.
Recently, a new mathematical framework for image reconstruction termed “compressed sensing” (CS) was formulated. In compressed sensing, only a small set of linear projections of a sparse image are required to reconstruct a quality image. The theory of CS is described by E. Candès, J. Romberg, and T. Tao, in “Robust uncertainty principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information,” IEEE Transactions on Information Theory 2006; 52:489-509, and by D. Donoho in “Compressed Sensing,” IEEE Transactions on Information Theory 2006; 52:1289-1306, and is disclosed, for example, in U.S. Pat. No. 7,646,924.
The work of Candès, et al., and Donoho produced both exact and approximate recovery sampling conditions for convex functionals, albeit with a higher sampling constant. In aspiration of reducing sampling requirements closer to the l0-associated theoretical limit, quasi-convex, almost everywhere (“a.e.”) differentiable prior functionals, such as P(v)=∥v∥pp with 0<p≦1 have been investigated recently. Such prior functionals more closely resemble the target l0-norm than does the l1-norm. While these methods cannot guarantee numerical achievement of global minima, they consistently outperform analogous convex techniques in practice.
The field of medical imaging can benefit from compressed sensing. In x-ray computed tomography (“CT”), fewer measurements translates to a lower radiation dose received by the patient. Similarly, in magnetic resonance imaging (“MRI”), decreasing the required number of measurements necessary to form an image allows for faster exams, which improves patient comfort and thereby minimizes the likelihood of subject motion. Additionally, clinical throughput is increased which potentially translates to reduced patient costs.
While natural and medical images share many characteristics, differences between the applications do exist and several key features unique to medical imaging have yet to be exploited within CS. In medical imaging, in general, image background is often defined by zero signal. Moreover, the spatial support of objects of interest in medical images is often known a priori or can be easily estimated. For example, in time-resolved HYPR imaging, as described, for example, by C. Mistretta, et al., in “Highly Constrained Backprojection for Time-Resolved MRI,” Magnetic Resonance in Medicine, 2006; 55(1):30-40, object support in a single time-frame is a subset of the support of the composite image used in the constrained backprojection operation. Support information could also be estimated from the low-resolution “scout” images typically acquired during pre-scan localization in both MRI and CT.
It would therefore be desirable to provide a method for image reconstruction that utilizes the framework of compressed sensing, while allowing a faster, more robust reconstruction process that is closer to an l0-minimization.